Abstract
We study the moduli space \(\overline{M}_{0,n}\) of genus 0 curves with n marked points. Following Keel and Tevelev, we give explicit polynomials in the Cox ring of \(\mathbb{P}^{1} \times \mathbb{P}^{2} \times \cdots \times \mathbb{P}^{n-3}\) that, conjecturally, determine \(\overline{M}_{0,n}\) as a subscheme. Using Macaulay2, we prove that these equations generate the ideal for 5 ≤ n ≤ 8. For n ≤ 6, we also give a cohomological proof that these polynomials realize \(\overline{M}_{0,n}\) as a subvariety of \(\mathbb{P}^{(n-2)!-1}\) embedded by the complete log canonical linear system.
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Acknowledgements
This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. We thank Bernd Sturmfels for providing inspiration and feedback on multiple drafts. We are also grateful to Christine Berkesch Zamaere, Renzo Cavalieri, Diane Maclagan, Steffen Marcus, Vic Reiner, and Jenia Tevelev for many helpful discussions. The second author was partially supported by a scholarship from the Clay Math Institute.
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Appendix
Appendix
We include the Macaulay2 code used to verify Conjecture 1.2.
R = QQ[a0,a1,b0,b1,b2,c0,c1,c2,c3,d0,d1,d2,d3,d4, e0,e1,e2,e3,e4,e5,f0,f1,f2,f3,f4,f5,f6];
The rows of the following matrix are coordinates on \(\mathbb{P}^{1}, \mathbb{P}^{2}, \mathbb{P}^{3}, \mathbb{P}^{4}, \mathbb{P}^{5}, \mathbb{P}^{6}\)
M = matrix{{0,0,0,0,0,0,0}, {a0,a1,0,0,0,0,0}, {b0,b1,b2,0,0,0,0}, {c0,c1,c2,c3,0,0,0}, {d0,d1,d2,d3,d4,0,0}, {e0,e1,e2,e3,e4,e5,0}, {f0,f1,f2,f3,f4,f5,f6} }; M05 = {{2,2}}; M06 = {{2,2},{3,2},{3,3}}; M07 = {{2,2},{3,2},{3,3},{4,2},{4,3},{4,4}}; M08 = {{2,2},{3,2},{3,3},{4,2},{4,3},{4,4},{5,2},{5,3}, {5,4},{5,5}}; M09 = {{2,2},{3,2},{3,3},{4,2},{4,3},{4,4},{5,2},{5,3}, {5,4},{5,5},{6,2},{6,3},{6,4},{6,5},{6,6}};
Select your desired n here:
L = M07;
Lemma 4.1 involves the 2 × 2-minors of the matrices
Q = apply(L, l -> {submatrix(M,{l_1-1,l_0}, 0..l_1-1), M_(l_1)_(l_0) } ) S = apply(Q, T -> matrix{ apply(entries transpose T_0, x -> x_0 * (x_1-T_1) ), (entries T_0)_1 })
We form the ideal J = J n of all such 2-minors, and compute the prime ideal I by saturation.
J = sum apply(S,N -> minors(2,N)); J = saturate(J,ideal(a0,a1)); J = saturate(J,ideal(b0,b1,b2)); J = saturate(J,ideal(c0,c1,c2,c3)); J = saturate(J,ideal(d0,d1,d2,d3,d4)); J = saturate(J,ideal(e0,e1,e2,e3,e4,e5)); I = saturate(J,ideal(f0,f1,f2,f3,f4,f5,f6));
The following are used to determine whether the dimension of I is correct, as well as compute the degree of I, and the minimal number of generators in each degree.
codim I, degree I, betti mingens I
We finally note that the initial ideal is square-free and Cohen–Macaulay :
M = monomialIdeal leadTerm I; betti mingens M
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Monin, L., Rana, J. (2017). Equations of \(\overline{M}_{0,n}\) . In: Smith, G., Sturmfels, B. (eds) Combinatorial Algebraic Geometry. Fields Institute Communications, vol 80. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7486-3_6
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